Heat Transfer

Heat Transfer

Heat is transmitted by three methods namely, Conduction, Convection and Radiation.

Conduction
It is the phenomenon of Heat transfer without the actual displacement of the particles of the medium. The particles of the medium execute vibratory motions
Ex. : Heat Transfer in a metal rod (solid)


Steady State : In the process of heat conduction through a conductor from hot end to cold end if no heat is absorbed by it along the conductor then it is called steady state of the conductor. The temperatures at different points of the conductor remain same.
(The temperature of each section is constant but not equal)
Under steady state of the conductor,
i) Rate of flow of heat = = constant
ii) Temperature gradient along the conductor = = constant (where )


3. Coefficient of Thermal Conductivity : K
The quantity of Heat conducted through a metal rod in steady state is
i) directly proportional to Area of cross section (A)
ii) directly proportional to temperature difference (₁ - ₂) between hot and cold ends
iii) directly proportional to time of flow of heat (t)
iv) inversely proportional to length (l) of the rod.


K is coefficient of Thermal Conductivity of the material of the conductor. It is property of the material of the conductor. It is independent of dimensions of the conductor.
(i) K of good conductor is determined by Searl's method
(ii) K of Insulator is determined by Lee's Disc method.


Junction Temperature is observed if two metal slabs of equal areas of cross-section, having lengths , , coefficients of thermal conductivities k₁,k₂and free and Temperatures q₁, q₂ are kept in contact with each other, then under steady state,
Junction temperature =
Special Case : If = = l then



Thermal Diffusivity (or) Thermometric conductivity D :
It is the ratio of coefficient of Thermal conductivity (K) to Thermal Capacity per unit volume (ms/v) of a material.




Thermal Conductance (C) : For a conductor ,
Thermal conductance =
Here
K = Coefficient of thermal conductivity
A = Area of cross section
= length of conductor
SI Unit : Watt (Kelvin)^-1
DF :


Thermal Resistance (R) :
i) Thermal resistance (R) of a conductor of length l, cross - section (A) and conductivity (K) is given by the formula
Thermal Resistance =
SI unit : Kw^-1



Convection
It is the phenomenon of Heat transfer by the actual displacement of the particles of the medium in a fluid.
Ex. : Heat Transfer in Liquids & Gases

Convection which results from difference in densities is called natural convection.
Ex: A fluid heated in a container.

If a heated fluid is forced to move by a blower (or) pump then the phenomenon is called forced convection [induced convection]
Ex: Temperature of human body is kept constant by pumping blood with heart pump. Here the transfer of heat is by forced convection.

The rate of heat convection from an object is such that

Here
A = Contact area

= Temperature difference between the object and conductive fluid.
h = constant called convection coefficient. It depends on the properties of the fluid such as density, viscosity, specific heat and thermal conductivity.


radiation
Radiation is the phenomenon of transfer of heat without necessity of a material medium. It is by virtue of electromagnetic waves.
Energy radiated from a body is called Radiant energy.
Rate of emission of radiant energy depends on
i) Nature of surface of the body
ii) Surface area of the body
iii) Temperature of the body and surroundings

Prevost's theory of Heat Exchange
i) Every body emits and absorbs heat radiations at all temperatures except at absolute zero (-273ºC)
ii) If a body emits more heat energy than what it absorbs from the surroundings, then its temperature falls.
iii) If a body absorbs more heat energy than what it emits then its temperature rises.
iv) If a body emits & absorbs heat in equal amounts, then it is said to be in Thermal equilibrium.
v) When the temperatures of body and surroundings are equalized, conduction and convection stop but the radiation exchange takes place.


Perfect blackbody
i) It is a body which absorbs all the heat radiations incident on it.
ii) On heating, it emits radiations of all possible wavelengths at a given temperature.
iii) The wavelengths of the emitted heat radiations depend only on the temperature but are independent of the material of the black body.
Ex : Lamp black (96%), platinum black (98%) Fery's and Wien's black bodies are artificial black bodies. 'Sun' is natural blackbody.


Spectral emissive power
It is the amount of energy radiated by unit surface area per second per unit wavelength range at a given temperature.

Emissive power depends upon Nature of the surface and temperature of the body.
It is maximum for a perfect black body . It is minimum for a smooth, shining white surface.


Emissivity or relative emittance (e) :
e =
For a perfect blackbody e = 1
For anybody 0 < e < 1
For a surface if a = Absorptive power,
r = Reflecting power, and t = Transmitting power then a + r + t =1
for a black body r =0 , t = 0, a=1


Kirchoff's law :
For a given temperature and wavelength, the ratio of emissive power to absorptive power of all bodies is always a constant. The constant is equal to emissive power of a perfect blackbody at the same temperature and wavelength.

i) Good emitters are good absorbers and vice versa.
ii) With increase of temperature increases.
the ratio also increases.
Dark lines in solar spectrum are called Fraunhoffer lines. Some wavelengths of white light from photosphere are absorbed by some elements in chromosphere
On the day of solar eclipse, absorption spectrum is not seen, rather emission spectrum which is complimentary to earlier absorption spectrum is seen.


Stefan's Law:
i) The amount of heat radiated per second from unit surface area of a black body (E) is proportional to Fourth Power of its absolute temperature (T).


where '' is Stefan's constant
= 5.67 × 10^-8w/m²/K⁴
ii) If the body is not a black body, then
= e s AT⁴('e' lies between 0 & 1)
e = Emissivity of the body.
Q = e At s T⁴


Stefan - Boltzmann Law :
i) If a blackbody at absolute temperature 'T_B' is in an enclosure at absolute temperature 'T_s' then the loss of thermal energy by the body per unit time is
= A (T_B⁴- T_S⁴)
ii) If it is not a blackbody, then
= e A (T_B⁴- T_S⁴) where e = emissivity


Newton's Law of cooling:
(i) The rate of cooling (the rate of fall of temperature) of a hot body is directly proportional to the difference between mean excess temperature of the body and the temperature of its surroundings

Here
_s = Temperature of surroundings.
The body is cooling from


Wien's Law:
i) Spectral energy density (E_l)
ii) Energy distribution can be explained by the formula

iii) This law is applicable to shorter wavelengths only (it is based upon classical mechanics)


Rayleigh - Jean's law :
i) Spectral Energy density El a l ^{- 4} (T) l ^{- 4}
ii) Energy distribution can be explained by the formula

K = 1.38 x 10^-23J/k/molecule
iii) This law is applicable to longer wavelengths only (it is based upon statistical mechanics)